IJMMS 26:5 (2001) 281–302
PII. S0161171201020117
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS
BETWEEN MINIMAL HOLOMORPHICALLY
NONDEGENERATE REAL ANALYTIC
HYPERSURFACES
JOËL MERKER
(Received 24 October 2000 and in revised form 19 April 2001)
Abstract. Recent advances in CR (Cauchy-Riemann) geometry have raised interesting fine
questions about the regularity of CR mappings between real analytic hypersurfaces. In analogy with the known optimal results about the algebraicity of holomorphic mappings between real algebraic sets, some statements about the optimal regularity of formal CR mappings between real analytic CR manifolds can be naturally conjectured. Concentrating on
the hypersurface case, we show in this paper that a formal invertible CR mapping between
two minimal holomorphically nondegenerate real analytic hypersurfaces in Cn is convergent. The necessity of holomorphic nondegeneracy was known previously. Our technique
is an adaptation of the inductional study of the jets of formal CR maps which was discovered by Baouendi-Ebenfelt-Rothschild. However, as the manifolds we consider are far from
being finitely nondegenerate, we must consider some new conjugate reflection identities
which appear to be crucial in the proof. The higher codimensional case will be studied in a
forthcoming paper.
2000 Mathematics Subject Classification. 32V25, 32V35, 32V40.
1. Introduction and statement of the results
1.1. Main theorem. Let (M, p) and (M , p ) be two small pieces of real analytic
hypersurfaces of Cn , with n ≥ 2. Here, the two points p ∈ M and p ∈ M are considered to be “central points.” Let t = (t1 , . . . , tn ) be some holomorphic coordinates
vanishing at p and let ρ(t, t̄) = 0 be a real analytic power series, a defining equation for (M, p). Similarly, we choose a defining equation ρ (t , t̄ ) = 0 for (M , p ).
Let h(t) = (h1 (t), . . . , hn (t)) be a collection of formal power series hj (t) ∈ C[[t]]
with hj (0) = 0. We will say that h induces a formal CR (Cauchy-Riemann) mapping
between (M, p) and (M , p ) if there exists a formal power series b(t, t̄) such that
ρ (h(t), h̄(t̄)) ≡ b(t, t̄)ρ(t, t̄). Further, h will be a formal equivalence between (M, p)
and (M , p ) if, in addition, the formal Jacobian determinant of h is nonzero, namely,
if det((∂hj /∂ti )(0))1≤i, j≤n ≠ 0. If the formal power series hj (t) are convergent, it follows from the identity ρ (h(t), h̄(t̄)) ≡ b(t, t̄)ρ(t, t̄) that h maps a neighborhood of p
in M biholomorphically onto a neighborhood of p in M . We are interested in optimal
sufficient conditions on the triple {M, M , h} which insure that the formal equivalence
h is convergent, namely, the series hj (t) converge for t small enough. To specify that
the mapping h is formal, we will write it “h : (M, p) →Ᏺ (M , p ),” with the index “Ᏺ”
referring to the word “formal.” The hypersurface (M, p) will be called minimal (at p,
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in the sense of Trépreau-Tumanov) if there does not exist a small piece of a complex (n − 1)-dimensional manifold passing through p which is contained in (M, p).
Recall also that (M , p ) is called holomorphically nondegenerate if there does not exist a nonzero (1, 0) vector field with holomorphic coefficients whose flow stabilizes
(M , p ). The present paper is essentially devoted to establish the following assertion.
Theorem 1.1. Let h : (M, p) →Ᏺ (M , p ) be a formal invertible CR mapping between
two real analytic hypersurfaces in Cn and assume that (M, p) is minimal. If (M , p ) is
holomorphically nondegenerate, then h is convergent.
(The reader is referred to the monograph [3] and to the articles [2, 4, 9, 13] for further background material.) This theorem provides a necessary and sufficient condition
for the convergence of an invertible formal CR map of hypersurfaces. The necessity
appears in a natural way (see Proposition 1.2 below). Geometrically, holomorphic nondegeneracy has a clear signification: it means that there exist no holomorphic tangent
vector field to (M , p ). This condition is equivalent to the nonexistence of a local complex analytic foliation of (Cn , p ) tangent to (M , p ). As matters stand, such a kind
of characterization for the regularity of CR maps happens to be known already in
case where at least one of the two hypersurfaces is algebraic (cf. [5, 6, 12]). In fact,
in the algebraic case, one can apply the classical “polynomial identities” in the spirit
of Baouendi-Jacobowitz-Treves. It was known that the true real analytic case requires
deeper investigations.
1.2. Brief history. Formal invertible CR mappings h : (M, p) →Ᏺ (M , p ) between
two local pieces of real analytic hypersurfaces in Cn have been proved to be convergent in various circumstances. Firstly, in 1974 by Chern-Moser, assuming that (M , p )
is Levi-nondegenerate. Secondly, in 1997 by Baouendi-Ebenfelt-Rothschild in [2], assuming that h is invertible (i.e., with nonzero Jacobian at p) and that (M , p ) is finitely
nondegenerate at p . And more recently in 1999, by Baouendi-Ebenfelt-Rothschild [4],
assuming for instance (but this work also contains other results) that (M , p ) is essentially finite, that (M, p) is minimal and that h is not totally degenerate, a result
which is valid in arbitrary codimensions. (Again, the reader may consult [3] for essential background on the subject, for definitions, concepts, and tools and also [9]
for related topics.) In summary, the above-mentioned results have all exhibited some
sufficient conditions.
1.3. Necessity. On the other hand, it is known (essentially since 1995, cf. [5]) that
holomorphic nondegeneracy of the hypersurface (M , p ) constitutes a natural necessary condition for h to be convergent, according to an important observation due
to Baouendi-Rothschild [2, 3, 5] (this observation followed naturally from the characterization by Stanton of the finite-dimensionality of the space of infinitesimal CR
automorphisms of (M, p) [16]; Stanton’s discovery is fundamental in the subject). We
may restate this observation as follows (see its proof at the end of Section 4).
Proposition 1.2. If (M , p ) is holomorphically degenerate, then there exists a nonconvergent formal invertible CR self map of (M , p ), which is simply of the form
Cn t Ᏺ exp( (t )L )(t ) ∈ Cn , where L is a nonzero holomorphic tangent vector
to (M , p ) and where the formal series (t ) ∈ C[[t ]], (0) = 0, is nonconvergent.
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
283
A geometric way to interpret this nonconvergent map would be to say that it is a map
which “slides in nonconvergent complex time” along the complex analytic foliation
induced by L , which is tangent to M by assumption. By this, we mean that each point
q of an arbitrary complex curve γ of the flow foliation induced by L is “pushed”
inside γ by means of a nonconvergent series corresponding to the time parameter
of the flow. This intuitive language can be illustrated adequately in the generic case
where the vector field L is nonzero at p . Indeed, we can suppose that L = ∂/∂t1
after a straightening and the above nonconvergent formal mapping is simply t Ᏺ
(t1 +(t ), t2 , . . . , tn
). Here, the t1 -lines are the leaves of the flow foliation of L and we
) by means of (t ) inside a leaf.
indeed “push” or “translate” the point (t1 , t2 , . . . , tn
Similar obstructions for the algebraic mapping problem stem from the existence
of complex analytic (or algebraic) foliations tangent to (M , p ) (cf. [5, 6]). Again,
this shows that the geometric notion of holomorphic nondegeneracy discovered by
Stanton is crucial in the field.
1.4. Jets of Segre varieties. The holomorphically nondegenerate hypersurfaces are
considerably more general and more difficult to handle than Levi-nondegenerate ones
[14, 15, 17], finitely nondegenerate ones [2], essentially finite ones [3, 4] or even
Segre nondegenerate ones [9]. The explanation becomes clear after a reinterpretation of these conditions in the spirit of the important geometric definition of jets
of Segre varieties due to Diederich-Webster [7]. In fact, these five distinct nondegeneracy conditions manifest themselves directly as nondegeneracy conditions of the
morphism of kth jets of Segre varieties attached to M , which is an invariant holomorphic map defined on its extrinsic complexification ᏹ = (M )c (we follow the
notations of Section 2). Here, the letter “c” stands for the “complexification operator.” In local holomorphic normal coordinates t = (w , z ) ∈ Cn−1 × C, vanishing
at p with τ := (ζ , ξ ) ∈ Cn−1 × C denoting the complexed coordinates (w , z )c ,
such that the holomorphic equation of the extrinsic complexification ᏹ is written
γ
ξ = z − iΘ (ζ , t ) = z − i γ∈Nn−1
ζ Θγ (t ) (cf. (2.3)), the conjugate complexified
∗
Segre variety is defined by ᏿t := {τ : ξ = z −iΘ (ζ , t )} (here, t is fixed; see [10] for
a complete exposition of the geometry of complexified Segre varieties) and the jet of
order k of the complex (n − 1)-dimensional manifold ᏿t at the point τ ∈ ᏿t defines
a holomorphic map
ϕk : ᏹ t , τ → jτk ᏿t ∈ Cn+Nn−1,k ,
Nn−1,k =
(n − 1 + k)!
,
(n − 1)!k!
(1.1)
given explicitly in terms of such a defining equation by a collection of power series
β
ϕk t , τ := jτk ᏿t = τ , ∂ζ ξ − z + iΘ ζ , t (1.2)
β∈Nn−1 , |β|≤k .
For k large enough, the various possible properties of this holomorphic map govern
some different nondegeneracy conditions on M which are appropriate for some generalizations of the Lewy-Pinchuk reflection principle. Let p c := (p , p̄ ) ∈ ᏹ . We give
here an account of five conditions, which can be understood as definitions.
(I) (M , p ) is Levi-nondegenerate at p if and only if ϕ1 is an immersion at p c .
(II) (M , p ) is finitely nondegenerate at p if and only if there exists k0 ∈ N∗ , ϕk is
an immersion at p c , for all k ≥ k0 .
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(III) (M , p ) is essentially finite at p if and only if there exists k0 ∈ N∗ , ϕk is a finite
holomorphic map at p c , for all k ≥ k0 .
(IV) (M , p ) is S-nondegenerate at p if and only if there exists k0 ∈ N∗ , ϕk |᏿ is
p̄
of generic rank dimC ᏿p̄ = n − 1, for all k ≥ k0 .
(V) (M , p ) is holomorphically nondegenerate at p if and only if there exists k0 ∈
N∗ , ϕk is of generic rank dimC ᏹ = 2n − 1, for all k ≥ k0 .
Remarks. (1) It follows from the biholomorphic invariance of Segre varieties that
two Segre morphisms of k-jets associated to two different local coordinates for (M , p )
are intertwined by a local biholomorphic map of Cn+Nn−1,k . Consequently, the properties of ϕk are invariant.
(2) The condition (I) is classical. The condition (II) is studied by Baouendi-EbenfeltRothschild [2, 3] and appeared already in Pinchuk’s thesis, in Diederich-Webster [7]
and in some of Han’s works. The condition (III) appears in Diederich-Webster [7] and
was studied by Baouendi-Jacobowitz-Treves and by Diederich-Fornaess. The condition
(IV) seems to be new and appears in [9]. The condition (V) was discovered by Stanton
in her concrete study of infinitesimal CR automorphisms of real analytic hypersurfaces (see [16] and the references therein) and is equivalent to the nonexistence of a
holomorphic vector field with holomorphic coefficients tangent to (M , p ). We claim
that (I)⇒(II)⇒(III)⇒(IV)⇒(V) (only the implication (IV)⇒(V) is not straightforward, see
Lemma 5.4 below for a proof). Finally, this progressive list of nondegeneracy conditions is the same, word by word, in higher codimensions.
1.5. A general commentary. To confirm evidence of the strong differences between these five levels of nondegeneracy, we point out some facts which are clear
at an intuitive and informal level. The immersive or finite local holomorphic maps
ϕ : (X, p) → (Y , q) between local pieces of complex manifolds with dimC X ≤ dimC Y
are very rare (from the point of view of complexity) in the set of maps of generic rank
equal to dimC X, or even in the set of maps having maximal generic rank m over a
submanifold (Z, p) ⊂ (X, p) of positive dimension m ≥ 1. Thus condition (V) is by
far the most general. Furthermore, an important difference between (V) and the other
conditions is that (V) is the only condition which is nonlocal, in the sense that it happens to be satisfied at every point if it is satisfied at a single point only, provided, of
course, that the local piece (M , p ) is connected. On the contrary, it is obvious that the
other four conditions are really local, even though they happen to be satisfied at one
point, there exist in general many other points where they fail to be satisfied. In this
concern, we recall that any (M , p ) satisfying (V) must satisfy (II) locally—hence also
(III) and (IV)—over a Zariski dense open subset of points of (M , p ) (this important
fact is proved in [3]). Therefore, the points satisfying (III) but not (II), or (IV) but not
(III), or (V) but not (IV), can appear to be more and more exceptional and rare from the
point of view of a point moving at random in (M , p ), but however, from the point of
view of local analytic geometry, which is the adequate viewpoint in this matter, they
are more and more generic and general, in truth.
Remark 1.3. An important feature of the theory of CR manifolds is to propagate the
properties of CR functions and CR maps along Segre chains, when (M, p) is minimal,
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
285
like iteration of jets [3], support of CR functions, and so forth. Based on this heuristic
idea, and believing that the generic rank of the Segre morphism over a Segre variety
is a propagating property, I have claimed in February 1999 (and provided a too quick
invalid proof) that any real analytic (M , p ) which is minimal at p happens to be
holomorphically nondegenerate if and only if it is Segre nondegenerate at p . This is
not true for a general (M , p ) as is shown, for instance, by an example from [4]. We
take in C3 equipped with the affine coordinates (z1 , z2 , z3 )
2 2 −1
−1
M : y3 = z1 1 + z1 z̄2 1 + Re z1 z̄2
− x3 Im z1 z̄2 1 + Re z1 z̄2
.
(1.3)
This algebraic hypersurface is holomorphically nondegenerate but is not Segre nondegenerate at the origin (use Lemmas 3.2 and 5.4 for a checking).
1.6. Summary of the proof. To the mapping h, we will associate the so-called in
variant reflection function ᏾h
(t, ν̄ ) as a C-valued map of (t, ν̄ ) ∈ (Cn , p) × (C̄n , p̄)
which is a series a priori only formal in t and holomorphic in ν̄ (the interest of studying the reflection function without any nondegeneracy condition on (M , p ) has been
pointed out for the first time by the author and Meylan in [11]). We prove in a first
step that ᏾h
and all its jets with respect to t converge on the first Segre chain. Then
using Artin’s approximation theorem [1] (the interest of this theorem of Artin for
the subject has been pointed out by Derridj in 1986, Séminaire sur les équations aux
dérivées partielles, Exposé no. XVI, Sur le prolongement d’applications holomorphes,
10pp., see page 5) and holomorphic nondegeneracy of (M , p ), we establish that the
formal CR map h converges on the second Segre chain. Finally, the minimality of (M, p)
together with a theorem of Gabrielov reproved elementarily by Eakin and Harris [8]
will both imply that h is convergent in a neighborhood of p. An important novelty is
the use of the conjugate reflection identities (5.5) below.
1.7. Closing remark. Two months after a first preliminary version of this paper
was finished (November 1999), distributed (January 2000) and then circulated as a
preprint, the author received in March 2000 a preprint (now published) [13] where
Theorems 1.1 and 9.1 were proved, using, in the first steps, an induction on the convergence of the mapping and its jets along Segre sets which was devised by BaouendiEbenfelt-Rothschild in [2]. But the proof that we provide here differs from the one in
[13] in the last step essentially. For our part, we introduce here in (5.5) and (8.2) a
crucial object which we call conjugate reflection identities. Essentially, this means that
both equivalent equations r (t , τ ) = 0 and r̄ (τ , t ) = 0 for (M , p ) (see Section 2.3)
must be considered and differentiated. More precisely, we mean that the CR derivations ᏸβ of Section 5.1 must be applied to (5.1), and to the conjugate of (5.1), which
yields (5.5). The author knows no previous paper where such an observation is done
and exploited. With this crucial remark at hand, the generalizations of Theorems 1.1
and 9.1 to higher codimensions can be performed completely, see the preprint Étude
de la convergence de l’application de symétrie CR formelle (in French), http://arxiv.org/
abs/math.CV/0005290, May 2000 (translated with the same proof in July 2000). The
first version of that preprint (http://arxiv.org/abs/math.CV/0005290v1) contained
some explicit hints in Section 18 for a second proof using conjugation of reflection
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identities and the last step of the proof given in [13]. The author believes that without
the use of the conjugation relation between r (t , τ ) and r̄ (τ , t ), no elementary
proof of Theorems 1.1 and 9.1 can be provided in higher codimension.
2. Preliminaries and notations
2.1. Defining equations. We will never speak of a germ. Thus, we will assume constantly that we are given two small local real analytic manifold-pieces (M, p) and
(M , p ) of hypersurfaces in Cn with centered points p ∈ M and p ∈ M . We first
choose local holomorphic coordinates t = (w, z) ∈ Cn−1 × C, z = x + iy and t =
(w , z ) ∈ Cn−1 × C, z = x + iy , vanishing at p and p such that the tangent spaces
to M and to M at 0 are given by {y = 0} and by {y = 0} in these coordinates. By this
choice, we carry out (cf. [3]) the equations of M and of M in the form
(2.1)
M : z = z̄ + iΘ̄ w, w̄, z̄ ,
M : z = z̄ + iΘ̄ w , w̄ , z̄ ,
where the power series Θ̄ and Θ̄ converge normally in (2r ∆)2n−1 for some small
r > 0. We denote by |t| := sup1≤i≤n |ti | the polydisc norm, so that (2r ∆)2n−1 =
{(w, ζ, ξ) : |w|, |ζ|, |ξ| < 2r }. Here, if we denote by τ := (t̄)c := (ζ, ξ) the extrinsic
complexification of the variable t̄, the equations of the complexified hypersurfaces
ᏹ := M c and ᏹ := (M )c are simply obtained by complexifying (2.1)
ᏹ : z = ξ + iΘ̄(w, ζ, ξ),
ᏹ : z = ξ + iΘ̄ w , ζ , ξ .
(2.2)
As in [3], we assume for convenience that the coordinates (w, z) and (w , z ) are
normal, that is, they are already straightened in order that Θ(ζ, 0, z) ≡ 0, Θ(0, w, z) ≡
0 and Θ (ζ , 0, z ) ≡ 0, Θ (0, w , z ) ≡ 0. This implies, in particular, that the Segre
varieties ᏿0 = {(w, 0) : |w| < 2r } and ᏿0 = {(w , 0) : |w | < 2r } are straightened to
the complex tangent plane to M at 0 and that, if we develop Θ̄ and Θ̄ with respect to
powers of w and w , then we can write
β
w β Θ̄β (ζ, ξ),
z = ξ + i
w Θ̄β ζ , ξ .
(2.3)
z = ξ +i
β∈Nn−1
∗
β∈Nn−1
∗
Here, we denote Nn−1
:= Nn−1 \{0}. So we mean that the two above sums begin with
∗
a w and w exponent of positive length |β| = β1 + · · · + βn−1 > 0. It is now natural
to set for notational convenience Θ̄0 (ζ, ξ) := ξ and Θ̄0 (ζ , ξ ) := ξ . Although normal
coordinates are in principle unnecessary, the reduction to such normal coordinates
will simplify a little the presentation of all our formal calculations below.
2.2. Complexification of the map. Now, the map h is by definition an n-vectorial
formal power series h(t) = (h1 (t), . . . , hn (t)) where hj (t) ∈ C[[t]], hj (0) = 0 and
det(∂hj /∂tk (0))1≤j, k≤n ≠ 0, which means that h is formally invertible. This map yields
by extrinsic complexification a map hc = hc (t, τ) = (h(t), h̄(τ)) between the two complexifications (ᏹ, 0) and (ᏹ , 0). In other words, if we denote h = (g, f ) ∈ Cn−1 × C in
accordance with the splitting of coordinates in the target space, the assumption that
hc (ᏹ) ⊂Ᏺ ᏹ reads as two equivalent fundamental equations

f (w, z) = f¯(ζ, ξ) + iΘ̄ g(w, z), ḡ(ζ, ξ), f¯(ζ, ξ)
ξ:=z−iΘ(ζ,w,z) ,
(2.4)
f¯(ζ, ξ) = f (w, z) − iΘ ḡ(ζ, ξ), g(w, z), f (w, z)
,
z:=ξ+iΘ̄(w,ζ,ξ)
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
287
after replacing ξ by z−iΘ(ζ, w, z) in the first line and z by ξ +iΘ̄(w, ζ, ξ) in the second
line. In fact, these (equivalent) identities must be interpreted as formal identities in
the rings of formal power series C[[ζ, w, z]] and C[[w, ζ, ξ]], respectively. Of course,
according to (2.2), we can equally choose the coordinates (ζ, w, z) or (w, ζ, ξ) over ᏹ.
In symbolic notation, we just write hc (ᏹ, 0) ⊂Ᏺ (ᏹ , 0) to mean the identities (2.4).
2.3. Conjugate equations, vector fields, and the reflection function. We also denote r (t, τ) := z−ξ −iΘ̄(w, ζ, ξ), r̄ (τ, t) := ξ −z+iΘ(ζ, w, z) and similarly r (t , τ ) :=
z−ξ −iΘ̄ (w , ζ , ξ ), r̄ (τ , t ) := ξ −z+iΘ (ζ , w , z ), so that ᏹ = {(t, τ) : r (t, τ) = 0},
ᏹ = {(t , τ ) : r (t , τ ) = 0} and the complexified Segre varieties are given by ᏿τp =
{(t, τp ) : r (t, τp ) = 0} ⊂ ᏹ for fixed τp , and ᏿tp = {(tp , τ) : r (tp , τ) = 0} ⊂ ᏹ for fixed tp
and similarly for ᏿τ , ᏿t (again, the reader is referred to [10] for a complete exp
p
position of the geometry of complexified Segre varieties). Finally, we introduce the
(n − 1) complexified (1, 0) and (0, 1) CR vector fields tangent to ᏹ, that we denote by
ᏸ = (ᏸ1 , . . . , ᏸn−1 ) and ᏸ = (ᏸ1 , . . . , ᏸn−1 ), and which can be given in symbolic vectorial
notation by
ᏸ=
∂
∂
+ iΘ̄w (w, ζ, ξ)
,
∂w
∂z
ᏸ=
∂
∂
− iΘζ (ζ, w, z)
.
∂ζ
∂ξ
(2.5)
The reflection function ᏾h
(t, ν̄ ), t ∈ Cn , ν̄ = (λ̄ , µ̄ ) ∈ Cn−1 ×C, will be, by definition,
the formal power series
β
᏾h t, ν̄ = ᏾h w, z, λ̄ , µ̄ = µ̄ − f (w, z) + i
λ̄ Θβ g(w, z), f (w, z) . (2.6)
β∈Nn−1
∗
We notice that this power series in fact belongs to the local “hybrid” ring C{ν̄ }[[t]].
3. Minimality and holomorphic nondegeneracy
3.1. Two characterizations. At first, we need to remind the two explicit characterizations of each one of the main two assumptions of Theorem 1.1. Let M be a real
analytic CR hypersurface given in normal coordinates (w, z) as above in (2.1).
Lemma 3.1 (see [3]). The following properties are equivalent:
(1) Θ̄(w, ζ, 0) ≡ 0.
(2) (∂ Θ̄/∂ζ)(w, ζ, 0) ≡ 0.
(3) M is minimal at 0.
(4) The Segre variety S0 is not contained in M.
(5) The holomorphic map C2n−2 (w, ζ) (w, iΘ̄(w, ζ, 0)) ∈ Cn has generic rank n.
Lemma 3.2 (see [2, 3, 16]). If the coordinates (w , z ) are normal as in (2.1), then the
real analytic hypersurface M is holomorphically nondegenerate at 0 if and only if there
n
exist β1 , . . . , βn−1 ∈ Nn−1
∗ , β := 0, such that
∂Θβ i ≡ 0 in C w , z .
w ,z
(3.1)
det
∂tj
1≤i, j≤n
Remark 3.3. Since for β = 0, we have Θβ (t ) = Θ0 (t ) = z , we see that (3.1) holds
if and only if det((∂Θβ i /∂wj )(w , z ))1≤i, j≤n−1 ≡ 0. Further, we can precise the other
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classical nondegeneracy conditions (I), (II), and (III) of Section 1 (for condition (IV),
see Lemma 5.4).
Lemma 3.4. The following concrete characterizations hold in normal coordinates.
(1) M is Levi nondegenerate at 0 if and only if the map w (Θβ (w , 0))|β|=1 is
immersive at 0.
(2) M is finitely nondegenerate at 0 if and only if there exists k0 ∈ N∗ such that the
map Cn−1 w (Θβ (w , 0))1≤|β|≤k0 is immersive at 0 for all k ≥ k0 .
(3) M is essentially finite at 0 if and only if there exists k0 ∈ N∗ such that the map
Cn−1 w (Θβ (w , 0))1≤|β|≤k0 is finite at 0 for all k ≥ k0 .
3.2. Switch of the assumptions. It is now easy to observe that the nondegeneracy
conditions upon M transfer to M through h and vice versa.
Lemma 3.5. Let h : (M, 0) →Ᏺ (M , 0) be a formal invertible CR map between two
real analytic hypersurfaces. Then
(1) (M, 0) is minimal if and only if (M , 0) is minimal.
(2) (M, 0) is holomorphically nondegenerate if and only if (M , 0) is holomorphically
nondegenerate.
Proof. We admit and use in the proof that minimality and holomorphic nondegeneracy are biholomorphically invariant properties. Let N ∈ N∗ be arbitrary. Since
h is invertible, after composing h with a biholomorphic and polynomial mapping
Φ : (M , 0) → (M , 0) which cancels low order terms in the Taylor series of h at the
origin, we can achieve that h(t) = t + O(|t|N ). Since the coordinates for (M , 0) may
be nonnormal, we must compose Φ ◦ h with a biholomorphism Ψ : (M , 0) → (M , 0)
which straightens the real analytic Levi-flat union of Segre varieties |x|≤r Sh(0,x)
into
the real hyperplane {y = 0} (this is how one constructs normal coordinates). One
can also verify that Ψ (t) = t + O(|t|N ). Then all terms of degree ≤ N in the power
series of Θ coincide with those of Θ. Each one of the two characterizing properties
(1) of Lemma 3.1 and (3.1) of Lemma 3.2, is therefore satisfied by Θ if and only if it is
satisfied by Θ .
4. Formal versus analytic
4.1. Approximation theorem. We collect here some useful statements from local
analytic geometry that we will repeatedly apply in the article. One of the essential
arguments in the proof of the main theorem (Theorem 1.1) rests on the existence of
analytic solutions arbitrarily close in the Krull topology to formal solutions of some
analytic equations, a fact which is known as Artin’s approximation theorem. Let m(w)
denote the maximal ideal of the local ring C[[w]] of formal power series in w ∈ Cn ,
n ∈ N∗ . Here is the first of our three fundamental tools, which will be used to get the
Cauchy estimates which show that the reflection function converges on the first Segre
chain (see Lemma 6.3).
Theorem 4.1 (see [1]). Let R(w, y) = 0, R = (R1 , . . . , RJ ), where w ∈ Cn , y ∈ Cm ,
Rj ∈ C{w, y}, Rj (0) = 0, be a converging system of holomorphic equations. Suppose
ĝ(w) = (ĝ1 (w), . . . , ĝm (w)), ĝk (w) ∈ C[[w]], ĝk (0) = 0, are formal power series which
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
289
solve R(w, ĝ(w)) ≡ 0 in C[[w]]. Then for every integer N ∈ N∗ , there exists a convergent series solution g(w) = (g1 (w), . . . , gm (w)), that is, satisfying R(w, g(w)) ≡ 0,
such that g(w) ≡ ĝ(w) (mod m(w)N ).
4.2. Formal implies convergent: first recipe. The second tool will be used to prove
that h is convergent on the second Segre chain, that is, h(w, iΘ̄(ζ, w, 0)) ∈ C{w, ζ}
(see Section 8).
Theorem 4.2. Let R(w, y) = 0, where R = (R1 , . . . , RJ ), w ∈ Cn , y ∈ Cm , Rj ∈ C{w, y},
Rj (0) = 0, be a system of holomorphic equations. Suppose that ĝ(w) = (ĝ1 (w), . . . , ĝm (w))
∈ C[[w]]m , ĝk (0) = 0 are formal power series solving R(w, ĝ(w)) ≡ 0 in C[[w]]. If
J ≥ m and if there exist j1 , . . . , jm with 1 ≤ j1 < j2 < · · · < jm ≤ J such that
det
∂Rjk ∂yl
w, ĝ(w)
≡ 0
in C[[w]],
(4.1)
1≤k, l≤m
then the formal power series ĝ(w) ∈ C{w} is in fact already convergent.
Remark 4.3. This theorem is a direct corollary of Artin’s theorem (Theorem 4.1).
The reader can find an elementary proof of it, for instance, in [9, Section 12].
4.3. Formal implies convergent: second recipe. The third statement will be applied to the canonical map of the second Segre chain, namely, to the map (w, ζ) (w, iΘ̄(ζ, w, 0)), which is of generic rank n by Lemma 3.1(5).
Theorem 4.4 (see [8]). Let a(y) ∈ C[[y]], y ∈ Cµ , a(0) = 0, be a formal power
µ
series and assume that there exists a local holomorphic map ϕ : (Cνx , 0) → (Cy , 0), of
maximal generic rank µ, that is, satisfying
∂ϕk
∃j1 , . . . , jµ , 1 ≤ j1 < · · · < jµ ≤ ν, s.t. det
(x)
≡ 0,
∂xjl
1≤k, l≤µ
(4.2)
and such that a(ϕ(x)) ∈ C{x} is convergent. Then a(y) ∈ C{y} is convergent.
4.4. Applications. We can now give an important application of Theorem 4.1,
namely, the Cauchy estimates for the convergence of the reflection function come
for free after one knows that all the formal power series Θβ (h(w, z)) ∈ C[[w, z]] are
convergent.
Lemma 4.5. Assume that h : (M, 0) →Ᏺ (M , 0) is a formal invertible CR mapping and
that M is holomorphically nondegenerate. Then the following properties are equivalent:
(1) h(w, z) ∈ C{w, z}n .
(2) ᏾h
(w, z, λ̄, µ̄) ∈ C{w, z, λ̄, µ̄}.
(3) Θβ (h(w, z)) ∈ C{w, z}, ∀β ∈ Nn−1 and ∃ε > 0 ∃C > 0 such that |Θβ (h(w, z))| ≤
C |β|+1 , for all (w, z) with |(w, z)| < ε and all β ∈ Nn−1 .
(4) Θβ (h(w, z)) ∈ C{w, z}, ∀β ∈ Nn−1 .
Proof. The implications (1)⇒(2)⇒(3)⇒(4) are straightforward. On the other hand,
consider the implication (4)⇒(1). By assumption, there exist convergent power series
ϕβ (w, z) ∈ C{w, z} such that Θβ (h(w, z)) ≡ ϕβ (w, z) in C[[w, z]]. It then follows
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JOËL MERKER
that h(t) is convergent by an application of Theorem 4.2 with Rn (t, t ) := z − ϕ0 (t)
and Ri (t, t ) := Θβ i (t ) − ϕβ i (t), 1 ≤ i ≤ n − 1 and where the multi-indices β1 , . . . , βn−1
are chosen as in Lemma 3.2 (use the property det((∂hj /∂tk )(0))1≤j, k≤n ≠ 0 and the
composition formula for Jacobian matrices to check that (4.1) holds).
Proof of Proposition 1.2. Let ϕ : (t , u ) exp(u L )(t ) = ϕ (t , u ) be the
n
local flow of the holomorphic vector field L = k=1 ak (t )∂/∂tk tangent to M . Of
course, this flow is holomorphic with respect to t ∈ Cn and u ∈ C, for |t |, |u | ≤ ε,
ε > 0. This flow satisfies ϕ (t , 0) ≡ t and ∂u ϕk (t , u ) ≡ ak (ϕ (t , u )). As L ≠ 0,
we have ∂u ϕ (t , u ) ≡ 0. We can assume that ∂u ϕ1 (t , u ) ≡ 0. Let (t ) ∈
C[[t ]]\C{t }, (0) = 0, be a nonconvergent formal power series which satisfies further ∂u ϕ1 (t , (t )) ≡ 0 in C[[t ]] (there exist many of such). If the formal power
series h7 : t Ᏺ ϕ (t , (t )) would be convergent, then t Ᏺ (t ) would also be
convergent, because of Theorem 4.2, contrarily to the choice of . Finally, L being
tangent to (M , 0), it is clear that h7 (M , 0) ⊂Ᏺ (M , 0).
5. Classical reflection identities
5.1. The fundamental identities. In this paragraph, we start the proof of our main
theorem (Theorem 1.1) by deriving the classical reflection identities. Thus let β ∈
Nn−1
∗ . By γ ≤ β, we mean γ1 ≤ β1 , . . . , γn−1 ≤ βn−1 . Denote |β| := β1 + · · · + βn−1 and
βn−1
β
ᏸβ := ᏸ1 1 · · · ᏸn−1
. Then applying all these derivations of any order (i.e., for each
n−1
) to the identity r̄ (h̄(τ), h(t)), that is, to
β∈N
f¯(ζ, ξ) ≡ f (w, z) − i
γ∈Nn−1
∗
ḡ(ζ, ξ)γ Θγ g(w, z), f (w, z) ,
(5.1)
as (w, z, ζ, ξ) ∈ ᏹ, it is well known that we obtain an infinite family of formal identities
that we recollect here in an independent technical statement (for the proof, see [3, 9]).
Lemma 5.1. Let h : (M, 0) →Ᏺ (M , 0) be a formal invertible CR mapping between
Ꮿ hypersurfaces in Cn . Then for every β ∈ Nn−1
∗ , there exists a collection of universal
polynomial uβ,γ , |γ| ≤ |β| in (n − 1)Nn−1,|β| variables, where Nk,l := (k + l)!/k! l! and
there exist holomorphic C-valued functions Ωβ in (2n − 1 + nNn,|β| ) variables near
ω
1
γ1
0 × 0 × 0 × (∂ξα ∂ζ h̄(0))|α1 |+|γ 1 |≤|β| in Cn−1 × Cn−1 × C × CnNn,|β| such that the identities
1 β ∂ Θ ḡ(ζ, ξ), g(w, z), f (w, z)
β! ζ
(β + γ)!
= Θβ g(w, z), f (w, z) +
ḡ(ζ, ξ)γ Θβ+γ
g(w, z), f (w, z)
β!γ!
γ∈Nn−1
∗
γ ¯
δ
ᏸ f (ζ, ξ)uβ,γ ᏸ ḡ(ζ, ξ) |δ|≤|β|
≡
∆(w, ζ, ξ)2|β|−1
|γ|≤|β|
1 1
γ
=: Ωβ w, ζ, ξ, ∂ξα ∂ζ h̄(ζ, ξ) 1
1
|α |+|γ |≤|β|
=: ωβ (w, ζ, ξ)
(5.2)
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
291
hold as formal power series in C[[w, ζ, ξ]], where
∆(w, ζ, ξ) = ∆(w, z, ζ, ξ)|z=ξ+iΘ̄(w,ζ,ξ) := det ᏸ ḡ
∂ ḡ
∂ ḡ
= det
(ζ, ξ) − iΘζ (ζ, w, z)
(ζ, ξ) .
z=ξ+iΘ̄(w,ζ,ξ)
∂ζ
∂ξ
(5.3)
Remark 5.2. The terms Ωβ , holomorphic in their variables, arise after writing
1
δ
γ1
ᏸ h̄(ζ, ξ) as χδ (w, z, ζ, ξ, (∂ξα ∂ζ h̄(ζ, ξ))|α1 |+|γ 1 |≤|δ| ) (by noticing that the coefficients
of ᏸ are analytic in (w, z, ζ, ξ) and by replacing again z by ξ + iΘ̄(w, ζ, ξ)).
5.2. Convergence over a uniform domain. From Lemma 5.1 which we have written
down in the most explicit way, we deduce the following useful observations. First, as
we have by the formal stabilization of Segre varieties h({w = 0}) ⊂Ᏺ {w = 0} and
as h is invertible, then it also holds that det(ᏸḡ(0)) = det(∂gj /∂wk (0))1≤j, k≤n−1 ≠ 0,
whence the rational term 1/∆2|β|−1 ∈ C[[w, ζ, ξ]] defines a true formal power series
at the origin. Putting now (ζ, ξ) = (0, 0) in (5.3) and shrinking r if necessary, we then
readily observe that ∆1−2|β| (w, 0, 0) ∈ ᏻ((r ∆)n−1 , C), since Θζ (0, w, 0) ∈ ᏻ((r ∆)n−1 , C)
γ1
and since the terms ∂ζ ḡ(0, 0) for |γ 1 | = 1 and ∂ξ1 ḡ(0, 0) are constants. Clearly, the
numerator in the middle identity (5.2) is also convergent in (r ∆)n−1 after putting
(ζ, ξ) = (0, 0), and we deduce finally the following important property
1 1
γ
(5.4)
Ωβ w, 0, 0, ∂ξα ∂ζ h̄(0, 0) 1
∈ ᏻ (r ∆)n−1 , C ,
1
|α |+|γ |≤|β|
for all β ∈ Nn−1
∗ . In other words, the domains of convergence of the ωβ (w, 0, 0) are
independent of β.
5.3. Conjugate reflection identities. On the other hand, applying the same derivations ᏸβ ’s to the conjugate identity r (h(t), h̄(τ)) = 0, we would get another family
of what we call conjugate reflection identities
0 ≡ ᏸβ f¯(ζ, ξ) + i
g(w, z)γ ᏸβ Θ̄γ ḡ(ζ, ξ), f¯(ζ, ξ) .
(5.5)
γ∈Nn−1
∗
The following lemma is the reason why these equations furnish essentially no more
information for the reflection principle.
Lemma 5.3. If (t, τ) ∈ ᏹ, then
ᏸβ r h(t), h̄(τ) = 0, ∀β ∈ Nn−1 ⇐⇒ ᏸβ r̄ h̄(τ), h(t) = 0, ∀β ∈ Nn−1 . (5.6)
Proof. As the two equations for ᏹ are equivalent, there exists an invertible formal
series α(t, τ) such that r (h(t), h̄(τ)) ≡ α(t, τ)r̄ (h̄(τ), h(t)). Thus
ᏸβ r h(t), h̄(τ) ≡ α(t, τ)ᏸβ r̄ h̄(τ), h(t)
β
αγ (t, τ)ᏸγ r̄ h̄(τ), h(t) ,
+
(5.7)
γ≤β, γ≠β
β
for some formal series αγ (t, τ) depending on the derivatives of α(t, τ). The implication “⇐” follows at once and the reverse implication is totally similar.
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JOËL MERKER
5.4. Heuristics. Nevertheless, in the last step of the proof of Theorem 1.1, equations (5.5) will be of crucial use, in place of (5.2) which will happen to be unusable. The
1
γ1
explanation is the following. Whereas the jets (∂ξα ∂ζ h̄(ζ, ξ))|α1 |+|γ 1 |≤|β| of the map-
ping h̄ cannot be seen directly to be convergent on the first Segre chain ᏿01 := {(w, 0)},
a convergence which would be a necessary fact to be able to use formula (5.2) again
in order to pass from the first to the second Segre chain ᏿20 := {(w, iΘ̄(w, ζ, 0))}
it will be possible—fortunately!—to show in Section 7 that the jets of the reflection
function ᏾h
itself converge on the first Segre chain, namely, that all the derivatives
β
ᏸ (Θ̄γ (ḡ(ζ, ξ), f¯(ζ, ξ))), restricted to the conjugate first Segre chain ᏿10 = {(ζ, 0)},
converge. In summary, we will only be able a priori to show that the jets of ᏾h
converge on the first Segre chain, and thus only (5.5) will be usable in the next step, but
not the classical reflection identities (5.2). This shows immediately why the conjugate
reflection identities (5.5) should be undertaken naturally in this context.
5.5. The Segre-nondegenerate case. Nonetheless, in the Segre-nondegenerate case,
which is less general than the holomorphically nondegenerate case, we have been able
to show directly that the jets of h converge on the first Segre chain (see [9]), and so on
by induction, without using conjugate reflection identities. The explanation is simple,
in the Segre nondegenerate case, we have first the following characterization, which
shows that we can separate the w variables from the z variable.
Lemma 5.4. The Ꮿω hypersurface M , given in normal coordinates (w , z ), is Segresuch that
nondegenerate at 0 if and only if there exist β1 , . . . , βn−1 ∈ Nn−1
∗
∂Θβi ≡ 0 in C w .
(5.8)
det
w ,0
∂wj
1≤i, j≤n−1
Also, M is holomorphically nondegenerate at 0 if it is Segre nondegenerate at 0.
Proof. In our normal coordinates, it follows that ᏿p = ᏿0 = {(w , 0, 0, 0)} and
w ({Θβ (w , 0)}|β|≤k ), whence the rephrasing (5.8) of definition (IV). As we
can take βn = 0 in (3.1), we see that the determinant of (3.1) does not vanish if (5.8)
holds. This proves the promised implication (IV)⇒(V).
ϕk |᏿0
Thanks to this characterization, we can delineate an analog to Lemma 4.5, whose
proof goes exactly the same way.
Lemma 5.5. Assume that h is invertible, that M is given in normal coordinates (2.1)
and that M is Segre nondegenerate. Then the following properties are equivalent:
(1) h(w, 0) ∈ C{w}.
(w, 0, λ̄, µ̄) ∈ C{w, λ̄, µ̄}.
(2) ᏾h
(3) Θβ (h(w, 0)) ∈ C{w}, ∀β ∈ Nn−1 and ∃ε > 0 ∃C > 0 such that |Θβ (h(w, 0))| ≤
C |β|+1 , ∀|w| < ε ∀β ∈ Nn−1 .
(4) Θβ (h(w, 0)) ∈ C{w}, ∀β ∈ Nn−1 .
5.6. Comment. In conclusion, in the Segre nondegenerate case (only) the convergence of all the components Θβ (h) of the reflection mapping after restriction to the
Segre variety ᏿0 = {(w, 0)} is equivalent to the convergence of all the components of h.
The same property holds for jets. Thus, in the Segre nondegenerate case, one can use
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
293
the classical reflection identities (5.2) (in which appear the jets of h̄, see Ωβ ) by induction on the Segre chains [9]. This is not so in the general holomorphically nondegenerate case, because it can happen that (3.1) holds whereas (5.8) does not hold, as
example (1.3) shows. In substance, one has therefore to use the conjugate reflection
identities. Now, the proof of Theorem 1.1 will be subdivided into three steps, which
will be achieved in Sections 6, 7, and 8.
6. Convergence of the reflection function on ᏿01
6.1. Examination of the reflection identities. The purpose of this paragraph is to
prove as a first step that the reflection function ᏾h
converges on the first Segre chain
᏿0 = {(w, 0)}.
Lemma 6.1. After perhaps shrinking the radius r > 0 of (5.4), the formal power series
᏾h (w, 0, λ̄ , µ̄ ) is holomorphic in (r ∆)n−1 × {0} × (r ∆)n .
Proof. We specify the infinite family of identities (5.1) (for β = 0) and (5.2) (for
n−1
β ∈ Nn−1
∗ ) on ᏿0 , to obtain first that f (w, 0) ≡ 0 ∈ C{w} and that for all β ∈ N∗
1 1
γ
Θβ g(w, 0), f (w, 0) ≡ Ωβ w, 0, 0, ∂ξα ∂ζ h̄(0, 0) 1
∈ C{w}.
(6.1)
1
|α |+|γ |≤|β|
Furthermore, since by (5.4) the Ωβ ’s converge for |w| < r and ζ = ξ = 0, we have
got Θβ (g(w, 0), f (w, 0)) ∈ ᏻ((r ∆)n−1 , C), for all β ∈ Nn−1 . It remains to establish a
Cauchy estimate like in Lemma 5.5(3). To this aim, we introduce some notation. We
set ϕ0 (w, z) := f (w, z) and ϕβ (w, z) := Θβ (g(w, z), f (w, z)) for all β ∈ Nn−1
∗ . By (6.1),
we already know that all the series ϕβ (w, 0) are holomorphic in {|w| < r }. Thus, in
order to prove that the reflection function restricted to the first Segre chain, namely
β
that the series = ᏾h
λ̄ ϕβ (w, 0) is convergent with respect to all
|᏿0 = µ̄ + i β∈Nn−1
∗
its variables, we must establish a crucial assertion.
Lemma 6.2. After perhaps shrinking r > 0, there exists a constant C > 0 with
ϕ (w, 0)
≤ C |β|+1 , ∀|w| < r , ∀β ∈ Nn−1 .
(6.2)
β
Proof. Actually, this Cauchy estimate will follow, by construction, from (6.1) and
from the property |Θβ (w , z )| ≤ C |β|+1 when (w , z ) satisfy |(w , z )| < r (the natural Cauchy estimate for Θ ), once we have proved the following independent and
important proposition, which is a rather direct application of Artin’s approximation
Theorem 4.1.
Lemma 6.3. Let w ∈ Cµ , µ ∈ N∗ , λ(w) ∈ C[[w]]ν , λ(0) = 0, ν ∈ N∗ , and let Ξβ (w, λ) ∈
C{w, λ}, Ξβ (0, 0) = 0, β ∈ Nm , m ∈ N∗ , be a collection of holomorphic functions
satisfying
(6.3)
∃r > 0 ∃ C > 0 s.t. Ξβ (w, λ)
≤ C |β|+1 , ∀β ∈ Nm , ∀
(w, λ)
< r .
Assume that Ξβ (w, λ(w)) ∈ ᏻ((r ∆)µ , C), for all β ∈ Nm and put Φβ (w) := Ξβ (w, λ(w)).
Then the following Cauchy inequalities are satisfied by the Φβ ’s
|β|+1
, ∀β ∈ Nm , ∀|w| < r1 .
(6.4)
∃0 < r1 ≤ r , ∃C1 > 0 s.t. Φβ (w)
≤ C1
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JOËL MERKER
Proof. We set Rβ (w, λ) := Ξβ (w, λ) − Φβ (w). Then Rβ ∈ ᏻ({|(w, λ)| < r }, C). By
noetherianity, we can assume that a finite subfamily (Rβ )|β|≤κ0 generates the ideal
(Rβ )β∈Nm , for some κ0 ∈ N∗ large enough. Applying now Theorem 4.1 to the collection of equations Rβ (w, λ) = 0, |β| ≤ κ0 , of which a formal solution λ(w) exists by
assumption, we get that there exists a convergent solution λ1 (w) ∈ C{w}ν vanishing
at the origin, that is, some λ1 (w) ∈ ᏻ((r1 ∆)µ , Cν ), for some 0 < r1 ≤ r , with λ1 (0) = 0,
which satisfies Rβ (w, λ1 (w)) ≡ 0, for all |β| ≤ κ0 . This implies that Rβ (w, λ1 (w)) ≡ 0,
for all β ∈ Nm . Now, we have obtained
Ξβ w, λ(w) ≡ Φβ (w) ≡ Ξβ w, λ1 (w) ,
∀β ∈ Nm .
(6.5)
The composition formula for analytic function then yields at once |Ξβ (w, λ1 (w))| ≤
|β|+1
C1
for |w| < r1 , after perhaps shrinking once more this positive number r1 in order
that |λ1 (w)| < r /2 if |w| < r1 . Thanks to (6.5), this gives the desired inequality for
Φβ (w). The Proof of Lemmas 6.1 and 6.2 are thus now complete.
7. Convergence of the jets of the reflection function on ᏿01
7.1. Transversal differentiation of the reflection identities. The next step in our
proof consists in showing that all the jets of the reflection function converge on the
first Segre chain ᏿0 .
Lemma 7.1. For all α ∈ N and all γ ∈ Nn−1 ,
γ
∂zα ∂w ᏾h
w, z, λ̄, µ̄ |z=0 ∈ C w, λ̄, µ̄ .
(7.1)
Equivalently, for all α ∈ N, for all γ ∈ Nn−1 , there exist r (α, γ) > 0 and C(α, γ) > 0
such that
α γ ∂ ∂w ϕ (w, z) z
β
z=0
|≤ C(α, γ)|β|+1
if |w| < r (α, γ), ∀β ∈ Nn−1
∗ .
(7.2)
Remark 7.2. Fortunately, the fact that r (α, γ) depends on α and γ will cause no
particular obstruction for the achievement of the last third step in Section 8. We believe however that this dependence should be avoided, but we get no immediate control of r (α, γ) as α + |γ| → ∞, in our proof—although it can be seen by induction that
γ
[∂zα ∂w ϕβ (w, z)]|z=0 ∈ ᏻ((r ∆)n−1 , C) (cf. the proof of Lemma 7.1).
Proof. If we denote by Ᏹα,γ the statement of the lemma, then it is clear that
Ᏹα,0 ⇒ Ᏹα,γ ∀γ ∈ Nn−1 .
(7.3)
It suffices therefore to establish the truth of Ᏹα,0 for all α ∈ N. We first establish that
∂zα |z=0 [ϕβ (w, z)] ∈ ᏻ((r ∆)n−1 , C), for all α ∈ N and all β ∈ Nn−1 . To this aim, we
specify the variables (w, z, ζ, ξ) := (w, z, 0, z) ∈ ᏹ (because Θ(0, w, z) ≡ 0) in (5.1) and
(5.2) to firstly obtain
f¯(0, z) ≡ f (w, z) − i
γ∈Nn−1
∗
ḡ(0, z)γ Θγ g(w, z), f (w, z)
(7.4)
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CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
and secondly the following infinite number of relations.
1 1
γ
Ωβ w, 0, z, ∂ξα ∂ζ h̄(0, z) 1
1
|α |+|γ |≤|β|
≡ Θβ g(w, z), f (w, z) +
γ∈Nn−1
∗
(β + γ)!
ḡ(0, z)γ Θβ+γ
g(w, z), f (w, z) .
β!γ!
(7.5)
Essentially, the game will consist in differentiating the equalities (7.4) and (7.5) with
respect to z at 0 up to arbitrary order α, in the aim to obtain new identities which
will yield ∂zα |z=0 [Θβ (g(w, z), f (w, z))] ∈ ᏻ((r ∆)n−1 , C), for all β ∈ Nn−1 and all α ∈ N,
by an induction process of “trigonal type.” We complete this informal description. To
begin with, for α = 1, after applying the derivation operator ∂z1 |z=0 to (7.4) and (7.5),
we get immediately
n−1
∂ ḡj (0, 0)Θγ g(w, 0), f (w, 0) ∈ C{w},
(7.6)
∂z1 f (w, 0) ≡ ∂z1 f¯(0, 0) + i
∂z
j=1
since ḡ(0, 0) = 0 (so ∂z1 [ḡ(0, z)γ ]z=0 = 0 for |γ| ≥ 2), and
1 1
γ
∂z1 Θβ g(w, 0), f (w, 0) = ∂z1 Ωβ w, 0, z, ∂ξα ∂ζ h̄(0, z)
|α1 |+|γ 1 |≤|β|
z=0
(β + γ)!
∂z1 ḡ(0, 0)γ Θβ+γ
−
g(w, 0), f (w, 0) ,
β!γ!
|γ|=1
(7.7)
making the slight abuse of notation ∂z1 χ(w, 0) instead of writing ∂z1 |z=0 [χ(w, z)] for
any formal power series χ(w, z) ∈ C[[w, z]]. For instance, ∂z1 ḡ(0, 0)γ signifies
n−1
[∂z1 (ḡ(0, z)γ )]|z=0 = k=1 γk ∂z1 ḡk (0, 0)[ḡ(0, 0)γ1 · · · ḡk (0, 0)γk −1 · · · ḡn−1 (0, 0)γn−1 ]. All
these expressions are convergent, because we already know (thanks to the first step)
(and even Θβ (g(w, 0), f (w, 0)) ∈
that Θβ (g(w, 0), f (w, 0)) ∈ C{w} for all β ∈ Nn−1
∗
ᏻ((r ∆)n−1 , C)) and because the derivative ∂z1 |z=0 (Ωβ ) can be expressed (thanks to the
1
1
chain rule) in terms of the derivatives ∂Ωβ /∂z, in terms of the derivatives ∂Ωβ /(∂ α ∂ γ )
(considering ∂
α1 γ 1
∂
as independent variables), and in terms of the derivatives
1 γ1
∂z1 |z=0 (∂ξα ∂ζ h̄(0, z)),
all taken at z = 0, which are terms obviously converging and
even which belong to the space ᏻ((r ∆)n−1 , C). Thus, we have got that ∂z1 ϕβ (w, 0) ∈
ᏻ((r ∆)n−1 , C), for all β ∈ Nn−1 (including β = 0). More generally, for arbitrary α ∈ N
α
¯
and β ∈ Nn−1
∗ , we observe readily that ∂z |z=0 [f (0, z)] is constant and, for the same
reasons as explained above, that
1
1
∂zα |z=0 Ωβ w, 0, z, ∂ξα ∂ζ γ h̄(0, z) 1
(7.8)
∈ ᏻ (r ∆)n−1 , C .
1
|α |+|γ |≤|β|
We can use this observation to perform a “trigonal” induction as follows. Let α0 ∈ N∗
and suppose by induction that ∂zα ϕβ (w, 0) ∈ ᏻ((r ∆)n−1 , C) for all α ≤ α0 and all
α +1
0
β ∈ Nn−1
|z=0 to (7.4), developing the expression
∗ . Then applying the derivation ∂z
α +1
according to Leibniz’ formula and using the fact that ∂z 0 |z=0 [ḡ(0, z)γ ] = 0 for all
|γ| ≥ α0 + 2, we get the expression
α +1
∂z 0
α +1
f (w, 0) ≡ ∂z 0
+i
f¯(0, 0)
α0 +1
0<|γ|≤α0 +1 κ=1
α +1 !
0
∂ κ ḡ(0, 0)γ ∂zα0 +1−κ ϕγ (w, 0).
κ! α0 + 1 − κ ! z
(7.9)
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JOËL MERKER
Now, thanks to the induction assumption and because the order of derivation in the
α +1−κ expression ∂z 0
ϕγ (w, 0) for 1 ≤ κ ≤ κ0 + 1 is less than or equal to α0 , we obtain
that this expression (7.9) belongs to ᏻ((r ∆)n−1 , C). Concerning the differentiation of
(7.7) with respect to z, we also get that the term
α +1
∂z 0
α +1
ϕβ (w, 0) ≡ ∂z 0
−
ωβ (w, 0, 0)


α0 +1
α0 +1 !
(β+γ)!  α0+1−κ κ
γ
∂ ḡ(0, 0) ∂z
ϕβ+γ (w, 0)
β!γ!
κ! α0 +1 − κ ! z
κ=1
+1
0<|γ|≤α0
(7.10)
α0 +1
belongs to ᏻ((r ∆)n−1 , C). Again, the important fact is that in the sum κ=1 , only
the derivations ∂zα ϕβ (w, 0) for 0 ≤ α ≤ α0 occur. In summary, we have shown that
∂z ϕβ (w, 0) is convergent for all α ∈ N.
Intermezzo. The induction process can be said to be of “trigonal type” because
we are dealing with the infinite collection of identities (5.2) which can be interpreted
as a linear system Y = AX, where X denotes the unknown (Θβ )β∈Nn−1 and A is an
infinite trigonal matrix, as shows an examination of (5.2). Further, when we consider
the jets, we still get a trigonal system. The main point is that after restriction to the
first Segre chain {ζ = ξ = 0}, this trigonal system becomes diagonal (or with only
finitely many nonzero elements after applying ∂zα ), but this crucial simplifying property
fails to be satisfied after passing to the next Segre chains. To be honest, we should
recognize that the proof we are conducting here unfortunately fails (for this reason)
to be generalizable to higher codimensions. However, an important natural idea will
appear during the course of the proof, namely, the appearance of the natural (and new)
conjugate reflection identities (5.5) which we will heavily use in Section 8. For reasons
of symmetry, we have naturally wondered whether they can be exploited more deeply.
A complete investigation is contained in our subsequent work on the subject (quoted
in Section 1.7).
End of proof of Lemma 7.1. It remains to show that there exist constants
r (α) > 0, C(α) > 0 such that the estimate (7.2) holds for (α, γ) = (α, 0) : |∂zα ϕβ (w, 0)| ≤
C(α)|β|+1 if |w| < r (α), ∀β ∈ Nn−1
∗ . To this aim, we apply Lemma 6.3 with the suitable functions and variables. First, it is clear that there exist universal polynomials (the
explicit formula in dimension one for the derivative of a composition (dn /dx n )(ψ ◦
φ(x)) = (ψ◦φ)(n) (x) is known as Faa di Bruno’s formula, (one of the favorite students
α1
α2
of Cauchy): (1/n!)(ψ ◦ φ)(n) (x) =
α1 +2α2 +···+nαn =n (1/α1 !α2 ! · · · αn !(1!) (2!)
αn
α1
α2
(n)
αn (α1 +α2 +···+αn )
· · · (n!) )×(φ (x)) (φ (x)) · · · (φ (x)) ψ
(φ(x))) such that the
following composite derivatives can be written
∗α ∂zα Θβ h(w, z) = Pα ∇∗α
(7.11)
z h(w, z) , ∇t Θβ h(w, z) ,
where the nα-tuple ∇∗α h(w, z) := ((∂zk h1 (w, z), . . . , ∂zk hn (w, z))1≤k≤α ) and the
β ((α + n)!/α!n!−1)-tuple ∇∗α
t Θβ (t ) := (∂t Θβ (t ))1≤|β|≤α . We now consider these polyα
α
α
nomials as holomorphic functions Gβ = Gβ (∇α
z h) of the n(α + 1) variables ∇z h =
k
k
((∂z h1 , . . . , ∂z hn )0≤k≤α ) which satisfy, using (7.11),
α α
α
∂zα Θβ h(w, z) = Gβα ∇α
,
(7.12)
z h(w, z) = Gβ ∇z h |∇α
z h:=∇z h(w,z)
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
1≤j≤n
297
1≤j≤n
k
α
k
where the nα-tuple ∇α
z h(w, z) = (∂z hj (w, z))0≤k≤α and ∇z h := (∂z hj )0≤k≤α are n(α+1)
independent variables as we have just said above. Obviously, these functions Gβα (∇α
z h)
|β|+1
α
h)|
≤
C(α)
if
|∇
h|
<
r
,
because
the
funcsatisfy an estimate of the form |Gβα (∇α
z
z
∗α |β|+1
if |t | < r ,
tions ∇∗α
t Θβ (t ) satisfy an estimate of the form |∇t Θβ (t )| ≤ C (α)
for some constants C (α) > 0, r > 0, and because we have Pα (∇α
z h, 0) ≡ 0. We already know that there exist holomorphic functions χβα (w) = ∂zα ϕβ (w, 0) in {|w| < r }
such that the following formal identity holds.
indexed by β ∈ Nn−1
∗
α
α Gβα ∇α
z h(w, 0) = ∂z ϕβ (w, 0) = χβ (w) in C[[w]].
(7.13)
Now, a direct application of Lemma 6.3 yields the desired estimate
α ∂ ϕ (w, 0)
≤ C(α)|β|+1
z
β
if |w| < r (α).
(7.14)
Thus, we have completed the proof of Lemma 7.1.
Important remark. When α → ∞, the number (n + 1)α of variables in ∇α
z h also
becomes infinite. Thus, at each step we apply Artin’s theorem in Lemma 6.3, the r (α)
may shrink and go to zero as α → ∞.
8. Convergence of the mapping
8.1. Jump to the second Segre chain. We now complete the final third step by
establishing that the power series h(t) is convergent in a neighborhood of 0. Let ᏿20 =
{exp w ᏸ(exp ζ ᏸ(0)) : |w| < r , |ζ| < r } be the second conjugate Segre chain [10], or
equivalently in our normal coordinates ᏿20 = {(w, iΘ̄(w, ζ, 0), ζ, 0) : |w| < r , |ζ| < r }.
We prove that the map hc is convergent on ᏿20 . More precisely,
Lemma 8.1. The formal power series h(w, iΘ̄(ζ, w, 0)) ∈ C{w, ζ}n is convergent.
From this lemma, we see now how to achieve the proof of our Theorem 1.1.
Corollary 8.2. Then the formal power series h(w, z) ∈ C{w, z}n is convergent.
Proof. We just apply Theorem 4.4, taking into account Lemma 3.1(5).
Proof of Lemma 8.1. Thus, we have to show that h(w, iΘ̄(ζ, w, 0)) ∈ C{w, ζ}.
To this aim, we consider the conjugate reflection identities (5.1) and (5.5) for various
after specifying them over ᏿20 , that is, after setting (w, z, ζ, ξ) :=
β ∈ Nn−1
∗
(w, iΘ̄(ζ, w, 0), ζ, 0) ∈ ᏹ, which we may write explicitly as follows
f¯(ζ, 0) ≡ f w, iΘ̄(w, ζ, 0) − i
0 ≡ ᏸβ f¯(ζ, ξ) ξ=0 + i
γ∈Nn−1
∗
γ∈Nn−1
∗
ḡ(ζ, 0)γ Θγ h w, iΘ̄(w, ζ, 0) ,
γ g w, iΘ̄(w, ζ, 0)
ᏸβ Θ̄γ h̄(ζ, ξ) ξ=0 ,
(8.1)
for all β ∈ Nn−1
∗ . Let now κ0 ∈ N∗ be an integer larger than the supremum of the
lengths of some multi-indices βi ’s, 1 ≤ i ≤ n − 1, satisfying the determinant property
stated in (3.1) of Lemma 3.2, that is, κ0 ≥ sup1≤i≤n−1 |βi |. According to Lemma 7.1, if
we consider (8.1) only for a finite number of β’s, say for |β| ≤ κ0 , there will exist a
298
JOËL MERKER
positive number r1 > 0 with r1 ≤ r and a constant C1 > 0 such that each power series
β
[ᏸβ (Θ̄γ (h̄(ζ, ξ)))]ξ=0 =: χγ (w, ζ) is holomorphic in the polydisc {|w|, |ζ| < r1 } and
β
|γ|+1
satisfies the Cauchy estimate |χγ (w, ζ)| ≤ C1
when |(w, ζ)| < r1 . We can now
represent (8.1) under the brief form
sβ w, ζ, h w, iΘ̄(w, ζ, 0) ≡ 0, |β| ≤ κ0 ,
(8.2)
where the holomorphic functions sβ = sβ (w, ζ, t ) are simply defined by replacing the
β
terms [ᏸβ (Θ̄γ (h̄(ζ, ξ)))]ξ=0 by χγ (w, ζ) in (8.1), so that the power series sβ converges
in the set {|w|, |ζ|, |t | < r1 }. The goal is now to apply Theorem 4.2 to the collection
of (8.2) in order to deduce that h(w, iΘ̄(w, ζ, 0)) ∈ C{w, ζ}.
Remark 8.3. As noted in the introduction, another (more powerful) idea would
be to apply the Artin approximation theorem (Theorem 4.1) to (8.2) to deduce the
existence of a converging solution H(w, ζ) and then to deduce that the reflection
function itself converges on the second Segre chain (which is in fact quite easy using
Lemma 5.3). This will be achieved in Section 9.
n
arising in Lemma 3.2. We set β = 0
First, we make a precise choice of the βi ∈ Nn−1
∗
i
satisfying
and, for 1 ≤ i ≤ n−1, let β be the infimum of all the multi-indices β ∈ Nn−1
∗
i+1
n
> · · · > β for the natural lexicographic order on Nn−1 , and such that an
β>β
(n − i + 1) × (n − i + 1) minor of the n × (n − i + 1) matrix
∂Θβ n ∂Θβ ∂Θβi+1 ᏹᏭ᐀β,βi+1 ,...,βn (t ) :=
(8.3)
t
t
·
·
·
t
∂tj
∂tj
∂tj
1≤j≤n
does not vanish identically as a holomorphic function of t ∈ Cn . We thus have
det((∂Θ i /∂tj )(t ))1≤i, j≤n ≡ 0 in C{t }. Concerning the choice of κ0 , we also require
that
β




∂Θ i 
β

κ0 ≥ inf k ∈ N : det
h w, iΘ̄(w, ζ, 0) 

∂tj

1≤i, j≤n



m(ζ)k C[[w, ζ]] ,
∈


(8.4)
where m(ζ) is the maximal ideal of C[[ζ]]. We can choose such a finite κ0 , because we
know already that the determinant in (8.4) does not vanish identically (this fact can
be easily checked, after looking at the composition formula for Jacobians, because, in
1
n
view of Lemma 3.2, the determinant (8.3) for (β , . . . , β ) does not vanish identically
and because the determinant det((∂hj /∂tk )(w, iΘ̄(w, ζ, 0)))1≤j, k≤n does not vanish
identically in view of the invertibility assumption on h and in view of the minimality
criterion Lemma 3.1(5)). Thus, after these choices are made, in order to finish the proof
by an application of Theorem 4.2, it will suffice to show the following lemma.
Lemma 8.4. There exist β1 , . . . , βn−1 , βn (= 0) ∈ Nn−1 with |βi | ≤ 2κ0 such that
∂sβi det
≡ 0 in C[[w, ζ]].
(8.5)
w,
ζ,
h
w,
i
Θ̄(w,
ζ,
0)
∂tj
1≤i, j≤n
Proof. To this aim, we introduce some new power series. We set
ᏸβ ḡ(ζ, ξ)γ Θγ t ,
Rβ w, z, ζ, t := ᏸβ f¯(ζ, ξ) + i
γ∈Nn−1
∗
(8.6)
299
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
for all β ∈ Nn−1 , after expanding with respect to (w, z, ζ, ξ) the power series appearing in ᏸβ (ḡ(ζ, ξ)γ ), ᏸβ f¯(ζ, ξ) and after replacing ξ by z − iΘ(ζ, w, z), and similarly,
we set
γ
w ᏸβ Θ̄γ h̄(ζ, ξ) ,
(8.7)
Sβ w, z, ζ, t := ᏸβ f¯(ζ, ξ) + i
γ∈Nn−1
∗
in coherence with the notation in (8.2) and finally also, we set
Tβ w, z, ζ, t := −ωβ (w, ζ, ξ) + Θβ t +
γ∈Nn−1
∗
(β + γ)!
ḡ(ζ, ξ)γ Θβ+γ
t .
β!γ!
(8.8)
We first remark that, by the very definition of sβ and of Sβ , we have
∂Sβ ∂sβ w, ζ, h w, iΘ̄(w, ζ, 0) ≡
w, iΘ̄(w, ζ, 0), ζ, h w, iΘ̄(w, ζ, 0)
∂tj
∂tj
(8.9)
as formal power series, for all β ∈ Nn−1 , 1 ≤ j ≤ n. Next, we establish a useful correspondence between the vanishing of the generic ranks of (Rβ )|β|≤2κ0 , (Sβ )|β|≤2κ0 and
(Tβ )|β|≤2κ0 .
Lemma 8.5. The following properties are equivalent.
(1) det((∂Rβi /∂tj )(w, z, ζ, h(w, z)))1≤i, j≤n ≡ 0, ∀β1 , . . . , βn , |β1 |, . . . , |βn | ≤ 2κ0 .
(2) det((∂Sβi /∂tj )(w, z, ζ, h(w, z)))1≤i, j≤n ≡ 0, ∀β1 , . . . , βn , |β1 |, . . . , |βn | ≤ 2κ0 .
(3) det((∂Tβi /∂tj )(w, z, ζ, h(w, z)))1≤i, j≤n ≡ 0, ∀β1 , . . . , βn , |β1 |, . . . , |βn | ≤ 2κ0 .
The proof of Lemma 8.5 will be given just below.
End of proof of Lemma 8.4. To finish the proof of Lemma 8.4, we assume by
contradiction that (8.5) is untrue, that is, Lemma 8.5(2) holds with z = iΘ̄(w, ζ, 0).
According to Lemma 8.5(3), we also have that the generic rank of the n×(2κ0 +n−1)!/
(2κ0 )!(n − 1)! matrix
ᏺ2κ0 (w, ζ) :=
∂Θβ h w, iΘ̄(w, ζ, 0)
∂tj
+
γ∈Nn−1
∗
|β|≤2κ0
∂Θβ+γ
(β + γ)!
γ
ḡ(ζ, 0)
h w, iΘ̄(w, ζ, 0)
β!γ!
∂tj
1≤j≤n
(8.10)
is strictly less than n. After making some obvious linear combinations between the
columns of ᏺ2κ0 with coefficients being formal power series in ζ which are polynomial
with respect to the ḡj (ζ, 0) ∈ m(ζ), 1 ≤ j ≤ n−1, we can reduce ᏺ2κ0 to the matrix of
same formal generic rank
0
ᏺ2κ
(w, ζ) :=
0
∂Θβ h w, iΘ̄(w, ζ, 0)
∂tj
+
|γ|≥2κ0
|β|≤2κ0
∂Θβ+γ
(β + γ)!
ḡ(ζ, 0)γ
h w, iΘ̄(w, ζ, 0)
β!γ!
∂tj
+1−|β|
.
1≤j≤n
(8.11)
300
JOËL MERKER
0
Now, taking the submatrix ᏺκ00 of ᏺ2κ
for which |β| ≤ κ0 , we see that we have reduced
0
0
ᏺκ0 to the simpler form
ᏺκ00 (w, ζ) ≡ ᏺκ10 (w, ζ) mod m(ζ)κ0 +1 ᏹatn×(κ0 +n−1)!/(κ0 )!(n−1)! C[[w, ζ]] ,
where
ᏺκ10 (w, ζ)
But

det 
:=

∂Θ i β
h w, iΘ̄(w, ζ, 0) 
∂tj
(8.12)
|β|≤κ0
∂Θβ h w, iΘ̄(w, ζ, 0)
∂tj
≡ 0
in C[[w, ζ]]
.
(8.13)
1≤j≤n
mod m(ζ)κ0 +1 ,
(8.14)
1≤i, j≤n
i
by the choice of the β ’s and of κ0 , which is the desired contradiction.
Proof of Lemma 8.5. The equivalence (1)(3) follows by an inspection of the
proof of Lemma 5.1; to pass from the system Rβ = 0, |β| ≤ 2κ0 , to the system Tβ = 0,
|β| ≤ 2κ0 , we have only use in the proof some linear combinations with coefficients in
C[[ζ, ξ]]. The equivalence (1)(2) is related with the substance of Lemma 5.3. Indeed,
in the relation r (t , τ ) ≡ α (t , τ )r̄ (τ , t ), with α (0, 0) = −1, insert first τ := h̄(τ)
to get r (t , h̄(τ)) ≡ α (t , h̄(τ))r̄ (h̄(τ), t ) and then differentiate by the operator ᏸβ
to obtain
β
ᏸβ r t , h̄(τ) ≡ α t , h̄(τ) ᏸβ r̄ h̄(τ), t +
α γ t , t, τ ᏸγ r̄ h̄(τ), t . (8.15)
γ≤β, γ≠β
In our notations, Rβ (w, z, ζ, t ) = ᏸβ r (t , h̄(τ)) and Sβ (w, z, ζ, t ) = ᏸβ r̄ (h̄(τ), t ),
after replacing ξ by z − iΘ(ζ, w, z). We deduce
∂Sβ ∂Rβ w, z, ζ, h(w, z) = α h(w, z), h̄(τ)
w, z, ζ, h(w, z)
∂tj
∂tj
∂Sγ β
α γ t, τ, h(w, z)
+
w, z, ζ, h(w, z) .
∂tj
γ≤β, γ≠β
(8.16)
Equation (8.16) shows that the terms (∂Rβ /∂tj )(w, z, ζ, h(w, z)) are trigonal linear
combinations of the terms (∂Sγ /∂tj )(w, z, ζ, h(w, z)), γ ≤ β, with nonzero diagonal
coefficients. This completes the proofs of Lemmas 8.5 and 8.1 and completes finally
our proof of Theorem 1.1.
Remark 8.6. Once we know that h(w, z) ∈ C{w, z}, we deduce that the reflection
function associated with the formal equivalence of Theorem 1.1 is convergent, that
is, ᏾h
(w, z, λ̄, µ̄) ∈ C{w, z, λ̄, µ̄}.
9. Convergence of the reflection function. Using the conjugate reflection identities (5.5), we observe that we may prove the following more general statement (see
[13], where the first proof was provided differently). Notice also that a second proof
of Theorem 1.1 can be derived from Theorem 9.1 by applying afterward Lemma 3.2
and Theorem 4.2.
CONVERGENCE OF FORMAL INVERTIBLE CR MAPPINGS . . .
301
Theorem 9.1. Let h : (M, p) →Ᏺ (M , p ) be a formal invertible CR mapping between
two real analytic hypersurfaces in Cn and assume that (M, p) is minimal. Then the
reflection mapping ᏾h
is convergent, that is,
᏾h t, ν̄ ∈ C t, ν̄ .
(9.1)
Proof. We come back to the conjugate reflection identities (5.5) and we put for
the arguments (w, z, ζ, ξ) := (w, iΘ̄(w, ζ, 0), ζ, 0). By Lemma 7.1, we know that all the
terms ᏸβ (Θ̄γ (h̄)) with these arguments are convergent power series in (w, ζ). The
same holds for the terms ᏸβ (f¯). We thus get (8.2) where the term h(w, iΘ̄(w, ζ, 0))
is only formal. Since the equations sβ (w, ζ, t ) are analytic, we can apply Artin’s approximation theorem. Consequently, there exists a convergent power series H(w, ζ) ∈
C{w, ζ}n satisfying
sβ w, ζ, H(w, ζ) ≡ 0, ∀β ∈ Nn−1 .
(9.2)
Here, we consider the complete list of equations sβ = 0 for all β. Then using calculations similar to (8.15), (8.16), namely, by applying the CR derivations ᏸβ to the identity
r (t , h̄(τ)) ≡ α (t , h̄(τ))r̄ (h̄(τ), t ), we deduce that H satisfies the first family of reflection identities, namely,
ᏸβ f¯ ≡ −i
γ∈Nn−1
∗
ᏸβ ḡ γ Θγ (H).
(9.3)
Here, the arguments of ᏸ and of h̄ are (w, iΘ̄(w, ζ, 0), ζ, 0) and the arguments of H
are (w, ζ). Applying the calculation of Lemma 5.1 to (9.3) and comparing to (5.2), we
deduce that for all β ∈ Nn−1 , we have
ωβ ≡ Θβ (h) +
γ∈Nn−1
∗
≡ Θβ (H) +
γ∈Nn−1
∗
(β + γ)! γ ḡ Θβ+γ (h)
β!γ!
(β + γ)! γ ḡ Θβ+γ (H).
β!γ!
(9.4)
Here, the terms ωβ are formal power series of (w, ζ) which depend on the jets of h̄, as
shown by Lemma 5.1. As the infinite system (9.4) is trigonal, it is formally invertible,
and here, for this precise system, the inverse is easy to compute and we get the simple
formula
(β + γ)! γ
Θβ (h) ≡ ωβ +
ḡ ωβ+γ ≡ Θβ (H).
(−1)γ
(9.5)
β!γ!
n−1
γ∈N∗
We deduce that
Θβ h w, iΘ̄(w, ζ, 0) ≡ Θβ H(w, ζ) ∈ C{w, ζ},
(9.6)
which shows that all the components of the reflection mapping are convergent on
the second Segre chain. It remains to apply the theorem of Gabrielov (Eakin-Harris) to
deduce that Θβ (h(t)) is convergent for all β. The final Cauchy estimate follows as in
Lemma 6.2. The proof of Theorem 9.1 is now complete.
302
JOËL MERKER
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Joël Merker: Laboratoire D’Analyse, Topologie et Probabilités, Centre de Mathématiques et D’Informatique, UMR 6632, 39 rue Joliot Curie, F-13453 Marseille Cedex 13,
France
E-mail address: merker@cmi.univ-mrs.fr